Phonon-mediated electron-electron interaction in real space

We have calculated the phonon-mediated interaction between two $s$-band electrons as a function of their relative separation in real space. We consider the two cases of acoustic phonons in a nonpolar crystal and of optic phonons in an ionic crystal, and present specific results for electrons at the bottom of the band, using parameters of aluminum and lithium chloride, respectively. In the acoustic-phonon case, even when the Coulomb repulsion between the two electrons is not included, the interaction has a wide repulsive core, which extends to a separation of about 200 \AA{} and afterwards oscillates between attractive and repulsive regions as the distance between the electrons increases. Classically, i.e., when the electrons are infinitely heavy, this potential is attractive when the electrons are less than a few angstroms apart and oscillates at greater distances. For ionic crystals one anticipates that the phonon-mediated interaction causes merely the optic-phonon screening of the Coulomb repulsion; i.e., $\frac{{e}^{2}}{{\ensuremath{\epsilon}}_{\ensuremath{\infty}}r}$ is converted to $\frac{{e}^{2}}{{\ensuremath{\epsilon}}_{0}r}$. However, we find that this mechanism produces a striking oscillation (versus $r$) about the expected result. Consequently, in LiCl, for example, the total potential---the direct Coulomb repulsion plus the phonon-mediated interaction---has deep, attractive potential wells, the first and largest of which has a depth of 28 meV, with its minimum occurring when the electrons are 33 \AA{} apart.

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